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The odd genius who showed that one infinity was greater than another

Illustration for article titled The odd genius who showed that one infinity was greater than another

This problem of infinity was pondered by Georg Cantor. What he concluded started him down a road that wound through infamy, through respectability, and wound up in theology. Find out more than anyone ever cared to consider about the infinite.


Imagine a thin line, almost a thread, stretching to infinity in both directions. It runs to the end of the universe. It is, in essence, infinite. Now look at the space all around it. That also runs to the end of the universe. It's also infinite. Both are infinite, yes, but are they the same? Isn't one infinity bigger than the other?

That's the question that Georg Cantor, a German mathematician who died shortly before World War I ended, grappled with throughout his life. Infinity was supposed to be an absolute number, especially in mathematics, where dividing infinity by a billion or multiplying it by a billion results, invariably and always, in infinity. Cantor thought about it and came up with aleph-nought, a 'number' that counts all the integers — whole numbers without fractions — that there are in existence. Aleph-nought has to be infinity, since there are an infinite quantity of whole numbers. But then what about real numbers? Real numbers include rational numbers, and irrational numbers (like the square root of five), and integers. This has to be a greater infinite number than all the other infinite numbers.

Illustration for article titled The odd genius who showed that one infinity was greater than another

This was when mathematicians started, metaphorically, booing and hissing when Cantor went by. The idea simply didn't make sense to them. Infinity was infinity. That was the end of it. Cantor called his various sets of different quantities of infinity 'transfinite' numbers — they are also known as cardinal numbers — and designated aleph-nought as the smallest transfinite number in existence. The controversy his views stirred up cost him an appointment at the University of Berlin. It also cost him his sanity on many different occasions. Throughout his later life he stayed in mental hospitals regularly.

When he felt his depression coming on, he often stopped pursuing mathematics. To fill his time, he had other obsessions. He was sure, for example, that Francis Bacon wrote Shakespeare's plays. He would study Elizabethan literature for months, looking for some connection between the two. He searched through the writings of both Bacon and Shakespeare, believing that he found mathematical codes and riddles that showed that Bacon was the author of both. In 1896 and 1897, Cantor published pamphlet after pamphlet about the idea, before moving on to something wilder.

Just after the turn of the century, Cantor, newly convalesced after another stay in an institution, asked that he be allowed to teach philosophy instead of mathematics. It was lucky for many philosophy students that he didn't. He based his ideas about God on set theory, and published works that said that any universe that didn't include infinite numbers showed a limit to God's power. Returning to his ideas about transfinite numbers, he named God the Absolute Infinite - the only concept so infinite that it could not be surpassed by other sets.


Cantor's life did not have a happy ending. The First World War put more stress on him and reduced the ability of those around him to care for his mental health. He eventually died in a sanatorium, his nation still at war. His ideas about infinity, and set theory, are still studied and honored today.

Top Image: Tumblr

Aleph-Nought Image: Wiki Commons

Via Science World, SJSU, Gap System.


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Let me see if I understand.

There are more irrational numbers than counting numbers. Here's an easy explanation that makes more sense to me than Cantor's:

If you had an infinite amount of time, you could count from 1 up to infinity (1, 2, ...). But you couldn't even get from 1 to 2 if you were counting the infinite numbers in between (2, 1.1, 1.01, 1.001, etc). You'd never even make it to 3.

Ok, that seems reasonable.